How do you pronounce "significantly"?

In American English: /sɪgˈnɪfɪkəntli/ (sig-NIF-ik-uhntl-ee). In British English: /sɪgnˈɪfɪkəntli/ (sign-IF-ik-uhntl-ee). In Canadian English: /sɪgˈnɪfɪkəntli/ (sig-NIF-ik-uhntl-ee). In Australian English: /sɪɡnˈɪfɪkəntli/ (sign-IF-ik-uhntl-ee).

Is "significantly" pronounced differently in American and British English?

Yes. In American English it's pronounced /sɪgˈnɪfɪkəntli/ (sig-NIF-ik-uhntl-ee), while in British English it's /sɪgnˈɪfɪkəntli/ (sign-IF-ik-uhntl-ee).